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1. Based on Kuratowski's definition of ordered pairs, prove that:
$$
(a, b) = (c, d) \iff (a = b \land c = d)
$$

According to Kuratowski's definition, $(x, y) = {{x}, {x, y}}$.

If $(a, b) = (c, d)$, then ${{a}, {a, b}} = {{c}, {c, d}}$

  1. if $a = b$,
     then ${{a}, {a, b}} = {{a}, {a, a}} = {{a}} = {{c}, {c, d}}$,
     then ${c} = {c, d} = {a}$,
     then $a = c = d$,
     then $b = a = d$
     then $a = c, b = d$

  2. if $a \neq b$
     a. if ${a} = {c, d}$,
        then $a = c = d$, ${{c}, {c, d}} = {{a}}$,
        then ${{a}, {a, b}} = {{a}}$,
        then $a = b$,
        which contradicts $a \neq b$.
     b. the same goes for ${c} = {a, b}$
     c. if ${a} = {c}$,
        then $a = c$,
        then ${{a}, {a, b}} = {{c}, {c, d}} = {{a}, {a, d}}$,
        then ${a, b} = {a, d}$,
        then $b = d$.

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